# Hall subgroup

In a finite $\pi$-divisible group there is a Hall $\pi$-subgroup (a Hall subgroup whose order is divisible only by the prime numbers in $\pi$ while the index is coprime to any number in $\pi$), and all Hall $\pi$-subgroups are conjugate. A finite solvable group has a Hall $\pi$-subgroup for any set $\pi$ of prime numbers. Every $\pi$-subgroup of a finite solvable group is contained in a Hall $\pi$-subgroup, and all Hall $\pi$-subgroups are conjugate. A normal Hall subgroup $H$ of a finite group $G$ always has a complement in $G$, that is, a subgroup $D$ such that $G=H\cdot D$ and such that $H\cap D$ is trivial; all complements to $H$ in $G$ are conjugate. If a group has a nilpotent Hall $\pi$-subgroup (cf. Nilpotent group), then all Hall $\pi$-subgroups are conjugate, and every $\pi$-subgroup is contained in some Hall $\pi$-subgroup. In general, a Hall subgroup does not have these properties. For example, the alternating group $A_5$ of order 60 has no Hall $\{2,5\}$-subgroup. In $A_5$ there is a Hall $\{2,3\}$-subgroup of order 12, but there is a subgroup of order 6 which does not lie in a Hall subgroup. Finally, in the simple group of order 168 the Hall $\{2,3\}$-subgroups are not conjugate.